5th Grade Illustrative Mathematics

# 5th Grade Illustrative Mathematics: A Comprehensive Guide

## Introduction: The Power of Problem-Based Learning

Welcome to the world of 5th Grade Illustrative Mathematics (IM), a revolutionary approach to math education that empowers students to become confident, capable problem-solvers. This curriculum is built on the belief that all learners can excel in mathematics when given the right tools and opportunities to explore, reason, and collaborate. [1] The IM program is a problem-based core curriculum designed to address content and practice standards to foster learning for all. [2]

At the heart of Illustrative Mathematics is a commitment to the **Standards for Mathematical Practice**, which describe the expertise that mathematics educators at all levels should seek to develop in their students. [3] These practices encourage students to make sense of problems, reason abstractly, construct viable arguments, and model with mathematics. A 2025 study by WestEd provided the first causal evidence of the IM curriculum’s effectiveness, finding that schools using IM significantly outperformed their peers, particularly in the middle grades. [4]

This course will guide you through the eight units of the 5th Grade Illustrative Mathematics curriculum, providing a comprehensive overview of the key concepts, learning objectives, and instructional strategies. We will also embed engaging YouTube videos to bring the lessons to life and provide additional support for students and educators.

## Unit 1: Finding Volume

This unit introduces students to the concept of volume as a measurable attribute of three-dimensional objects. Through hands-on activities, students build their understanding of volume by packing rectangular prisms with unit cubes. This tactile experience helps them develop a strong conceptual foundation before moving on to more abstract formulas. The Illustrative Mathematics curriculum emphasizes a problem-based approach, so students will encounter real-world scenarios where they need to determine the volume of various objects. [5]

### From Concrete to Abstract: The Learning Progression

The learning progression in this unit is carefully designed to move students from concrete understanding to abstract reasoning. Initially, students will work with physical unit cubes to build and measure the volume of small rectangular prisms. They will then transition to pictorial representations, where they will need to visualize the layers of cubes within a prism. This progression helps students develop spatial reasoning skills and a deep understanding of the structure of rectangular prisms.

### Key Concepts and Learning Goals

| Concept | Description | Learning Goals |
| :— | :— | :— |
| **Volume as a Measurement** | Understanding that volume is the amount of three-dimensional space an object occupies and can be measured in cubic units. | Describe volume as the space taken up by a solid object. |
| **Unit Cubes** | Using unit cubes as a tool to measure volume and build rectangular prisms. | Measure the volume of a rectangular prism by finding the number of unit cubes needed to fill it. |
| **Volume Formula** | Developing and applying the formulas for the volume of a rectangular prism: `Volume = length × width × height` and `Volume = area of base × height`. | Use the layered structure in a rectangular prism to find volume. |
| **Additive Volume** | Finding the volume of composite figures by adding the volumes of the non-overlapping rectangular prisms that compose them. | Find the volume of a figure composed of rectangular prisms. |

### Problem-Based Learning in Action

A typical problem in this unit might ask students to determine how many small boxes can fit into a larger shipping container. This type of problem encourages students to think about volume in a practical context and to develop their own strategies for solving the problem. The teacher’s role is to facilitate the learning process by asking probing questions and encouraging students to share their thinking with their peers. This collaborative approach to problem-solving is a hallmark of the Illustrative Mathematics curriculum and has been shown to be highly effective in promoting deep conceptual understanding. [6]

## Unit 2: Fractions as Quotients and Fraction Multiplication

This unit builds on students’ prior knowledge of fractions and division, helping them to see the deep connection between these two concepts. A key understanding in this unit is that a fraction can be interpreted as the division of the numerator by the denominator. This powerful idea unlocks new ways of thinking about fractions and solving problems. The Illustrative Mathematics curriculum uses a variety of models and contexts to help students make sense of this concept, such as sharing problems and measurement scenarios. [5]

### Visual Models for Deep Understanding

Visual models play a crucial role in this unit, helping students to develop a conceptual understanding of fraction multiplication before they are introduced to the standard algorithm. Students will use area models, number lines, and other representations to visualize what it means to multiply a fraction by a whole number and a fraction by a fraction. These models provide a concrete foundation for the more abstract procedures that will be introduced later.

### Key Concepts and Learning Goals

| Concept | Description | Learning Goals |
| :— | :— | :— |
| **Fractions as Division** | Understanding that a fraction `a/b` can be interpreted as the result of dividing `a` by `b`. | Interpret a fraction as division of the numerator by the denominator. |
| **Multiplying a Fraction by a Whole Number** | Applying the concept of repeated addition and using visual models to multiply a fraction by a whole number. | Solve problems involving multiplication of a fraction by a whole number. |
| **Multiplying a Fraction by a Fraction** | Developing a conceptual understanding of fraction multiplication using area models and other visual representations. | Explain what it means to multiply a fraction by a fraction. |

### Connecting to Real-World Contexts

The problems in this unit are grounded in real-world contexts that are meaningful to students. For example, students might be asked to solve a problem about sharing a pizza among a group of friends or calculating the amount of ingredients needed for a recipe. These contexts help students to see the relevance of fractions in their daily lives and to develop a deeper understanding of the mathematical concepts.

## Unit 3: Multiplying and Dividing Fractions

In this unit, students transition from a conceptual understanding of fraction multiplication and division to procedural fluency. They will solidify their understanding of the standard algorithms for these operations and apply them to solve a wide range of problems, including those involving mixed numbers. The Illustrative Mathematics curriculum continues to emphasize sense-making, so students will be encouraged to check the reasonableness of their answers and to connect the procedures to the underlying concepts. [5]

### From Algorithms to Applications

While this unit focuses on developing procedural fluency, it does not do so in isolation. Students will apply their skills to solve real-world problems that require them to multiply and divide fractions. For example, they might be asked to calculate the area of a room with fractional dimensions or to determine how many servings of a certain size can be made from a given amount of food. These applications help students to see the practical value of fraction operations and to develop a deeper understanding of their meaning.

### Key Concepts and Learning Goals

| Concept | Description | Learning Goals |
| :— | :— | :— |
| **Multiplying Fractions Fluently** | Developing and applying the standard algorithm for multiplying fractions, including mixed numbers. | Fluently multiply fractions, including mixed numbers. |
| **Dividing a Whole Number by a Unit Fraction** | Understanding that dividing a whole number by a unit fraction is equivalent to multiplying the whole number by the reciprocal of the fraction. | Solve problems involving division of a whole number by a unit fraction. |
| **Dividing a Unit Fraction by a Whole Number** | Developing a conceptual understanding of dividing a unit fraction by a whole number and applying it to solve problems. | Solve problems involving division of a unit fraction by a whole number. |

### Addressing Common Misconceptions

This unit also provides opportunities to address common misconceptions about fraction operations. For example, many students struggle with the idea that dividing by a fraction can result in a larger number. The Illustrative Mathematics curriculum uses a variety of strategies to help students overcome these misconceptions, such as using visual models and connecting division to multiplication.

## Unit 4: Wrapping Up Multiplication and Division with Multi-digit Numbers

This unit serves as a crucial consolidation point, allowing students to deepen their understanding and achieve fluency with multiplication and division of multi-digit whole numbers. Building on work from previous grades, students will refine their use of the standard algorithms for these operations. The Illustrative Mathematics curriculum ensures that this practice is not rote memorization but is instead grounded in place value understanding and the properties of operations. [5]

### Fluency and Flexibility

The goal of this unit is not just to teach students how to follow a set of steps, but to help them develop a flexible and efficient approach to computation. Students will be encouraged to use a variety of strategies, including mental math, estimation, and the standard algorithms, to solve problems. They will also learn to check the reasonableness of their answers and to choose the most appropriate strategy for a given problem.

### Key Concepts and Learning Goals

| Concept | Description | Learning Goals |
| :— | :— | :— |
| **Multiplication of Multi-digit Numbers** | Applying the standard algorithm to multiply multi-digit whole numbers. | Fluently multiply multi-digit whole numbers using the standard algorithm. |
| **Division of Multi-digit Numbers** | Applying the standard algorithm to divide multi-digit whole numbers. | Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors. |
| **Problem-Solving with Multi-digit Operations** | Solving multi-step real-world problems involving multiplication and division of multi-digit numbers. | Solve real-world problems involving multiplication and division of multi-digit numbers. |

### Real-World Applications

To make the learning meaningful, this unit is rich with real-world applications. Students will solve problems involving topics such as budgeting, measurement, and data analysis. These problems will require them to not only perform the calculations correctly but also to interpret the results in the context of the problem. This focus on application helps students to see the power and utility of mathematics in their lives.

## Unit 5: Place Value Patterns and Decimal Operations

This unit extends students’ understanding of the place value system to include decimal numbers. Students will learn that the place value patterns they observed with whole numbers also apply to the right of the decimal point. They will learn to read, write, compare, and round decimals to the thousandths place. The Illustrative Mathematics curriculum uses a variety of representations, including base-ten blocks, number lines, and place value charts, to help students develop a deep understanding of decimal concepts. [5]

### The Power of Ten

A central theme in this unit is the power of ten. Students will explore how the value of a digit changes as it moves to the left or right in a number. They will learn that a digit in one place represents ten times as much as it represents in the place to its right and one-tenth of what it represents in the place to its left. This understanding is crucial for developing fluency with decimal operations.

### Key Concepts and Learning Goals

| Concept | Description | Learning Goals |
| :— | :— | :— |
| **Decimal Place Value** | Understanding the value of each digit in a decimal number to the thousandths place. | Recognize that a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. |
| **Reading and Writing Decimals** | Reading and writing decimals to the thousandths place in standard, word, and expanded form. | Read, write, and compare decimals to thousandths. |
| **Comparing and Ordering Decimals** | Using place value to compare and order decimals to the thousandths place. | Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. | | **Rounding Decimals** | Using place value understanding to round decimals to any place. | Use place value understanding to round decimals to any place. | | **Adding and Subtracting Decimals** | Applying the standard algorithms for addition and subtraction to decimal numbers. | Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. | ### Real-World Connections This unit is full of opportunities to connect decimal concepts to the real world. Students will work with money, measurement, and other contexts where decimals are used. For example, they might be asked to calculate the total cost of several items at a store or to compare the heights of different plants. These real-world connections help students to see the practical importance of decimals and to develop a deeper understanding of their meaning. ## Unit 6: More Decimal and Fraction Operations This unit brings together students' understanding of fractions and decimals, helping them to see the connections between these two representations of numbers. Students will learn to multiply and divide decimals and fractions, and they will apply these skills to solve a variety of multi-step problems. The Illustrative Mathematics curriculum emphasizes the use of visual models and real-world contexts to help students make sense of these complex operations. [5] ### Connecting Fractions and Decimals A key focus of this unit is on helping students to see that fractions and decimals are simply different ways of representing the same number. Students will learn to convert between fractions and decimals and to choose the most appropriate representation for a given problem. This flexibility is essential for developing a deep understanding of the number system. ### Key Concepts and Learning Goals | Concept | Description | Learning Goals | | :--- | :--- | :--- | | **Multiplying Decimals** | Applying strategies based on place value and the properties of operations to multiply decimals. | Multiply decimals to hundredths, using concrete models or drawings and strategies based on place value. | | **Dividing Decimals** | Applying strategies based on place value and the relationship between multiplication and division to divide decimals. | Divide decimals to hundredths, using concrete models or drawings and strategies based on place value. | | **Solving Problems with Fractions and Decimals** | Solving multi-step real-world problems involving operations with fractions and decimals. | Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. | ### Problem-Solving and Reasoning This unit is rich with opportunities for students to develop their problem-solving and reasoning skills. They will be asked to solve complex, multi-step problems that require them to apply their knowledge of fractions and decimals in a variety of contexts. They will also be asked to explain their thinking and to justify their answers, which helps them to develop a deeper understanding of the mathematical concepts. ## Unit 7: Shapes on the Coordinate Plane This unit introduces students to the coordinate plane, a foundational concept in mathematics that connects geometry and algebra. Students will learn to plot points in the first quadrant of the coordinate plane and to use coordinates to describe the location of points and the properties of shapes. The Illustrative Mathematics curriculum provides a playful and engaging introduction to this topic, with activities that involve creating and analyzing patterns and shapes on the coordinate plane. [5] ### A New Way of Seeing Shapes The coordinate plane provides a powerful new tool for analyzing and understanding geometric shapes. By representing shapes on the coordinate plane, students can use coordinates to describe their properties, such as side lengths, and to reason about their relationships. This unit lays the groundwork for more advanced topics in geometry and algebra that students will encounter in later grades. ### Key Concepts and Learning Goals | Concept | Description | Learning Goals | | :--- | :--- | :--- | | **The Coordinate Plane** | Understanding the structure of the coordinate plane and how to plot points in the first quadrant. | Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. | | **Plotting Points** | Plotting points in the first quadrant of the coordinate plane given their coordinates. | Represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. | | **Analyzing Shapes** | Using coordinates to analyze the properties of shapes on the coordinate plane. | Classify two-dimensional figures in a hierarchy based on properties. | ### From Patterns to Properties This unit begins with an exploration of patterns on the coordinate plane. Students will generate and graph numerical patterns and then analyze the relationships between the patterns. This work helps them to develop an understanding of the structure of the coordinate plane and to see the connection between numbers and shapes. They will then move on to plotting and analyzing geometric shapes, using coordinates to describe their properties and to classify them. ## Unit 8: Putting it All Together This capstone unit provides a rich opportunity for students to synthesize and apply the mathematical knowledge and skills they have developed throughout the year. The problems in this unit are designed to be challenging and engaging, requiring students to draw on their understanding of multiple concepts to arrive at a solution. The Illustrative Mathematics curriculum emphasizes the importance of perseverance and collaboration, and this unit provides a perfect context for students to practice these essential skills. [5] ### A Culminating Challenge This unit is not about learning new content, but about applying existing knowledge in new and creative ways. Students will work on multi-step problems that may involve fractions, decimals, volume, and the coordinate plane. These problems are designed to be open-ended, with multiple entry points and solution paths. This allows all students to engage with the mathematics at their own level and to experience the joy of mathematical discovery. ### Key Concepts and Learning Goals | Concept | Description | Learning Goals | | :--- | :--- | :--- | | **Problem-Solving** | Applying a variety of strategies to solve multi-step problems. | Make sense of problems and persevere in solving them. | | **Mathematical Modeling** | Using mathematics to model and solve real-world problems. | Model with mathematics. | | **Communication and Reasoning** | Explaining and justifying mathematical thinking. | Construct viable arguments and critique the reasoning of others. | ### A Celebration of Learning This unit can be seen as a celebration of all that students have learned in 5th grade mathematics. It is a chance for them to showcase their skills and to see how the different mathematical concepts they have studied are interconnected. By the end of this unit, students will have a deeper appreciation for the power and beauty of mathematics, and they will be well-prepared for the challenges of middle school mathematics. ## References [1] Illustrative Mathematics. (n.d.). *Math Curriculum*. Retrieved from https://illustrativemathematics.org/math-curriculum/ [2] Kendall Hunt Publishing. (n.d.). *IM K–5 Math*. Retrieved from https://im.kendallhunt.com/k5/curriculum.html [3] Common Core State Standards Initiative. (n.d.). *Standards for Mathematical Practice*. Retrieved from https://thecorestandards.org/Math/Practice/ [4] Khanani, N., Zabala, D., Stackhouse, S., Gu, J., & Walters, K. (2025). *Impact of the Illustrative Math Curriculum on Math Achievement*. WestEd. Retrieved from https://www.wested.org/resource/impact-of-the-illustrative-math-curriculum-on-math-achievement/ [5] Illustrative Mathematics. (n.d.). *Grade 5 Scope and Sequence*. Retrieved from https://curriculum.illustrativemathematics.org/k5/teachers/grade-5/course-guide/scope-and-sequence.html [6] ResearchGate. (n.d.). *Improving Fifth-Grade Students' Mathematical Problem-Solving and Critical Thinking Skills Using Problem-Based Learning*. Retrieved from https://www.researchgate.net/publication/341288800_Improving_Fifth-Grade_Students'_Mathematical_Problem-Solving_and_Critical_Thinking_Skills_Using_Problem-Based_Learning